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Theorem cotr3 14925
Description: Two ways of saying a relation is transitive. (Contributed by RP, 22-Mar-2020.)
Hypotheses
Ref Expression
cotr3.a 𝐴 = dom 𝑅
cotr3.b 𝐵 = (𝐴𝐶)
cotr3.c 𝐶 = ran 𝑅
Assertion
Ref Expression
cotr3 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝐴𝑦𝐵𝑧𝐶 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝐴,𝑦,𝑧   𝑦,𝐵,𝑧   𝑧,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem cotr3
StepHypRef Expression
1 cotr3.a . . 3 𝐴 = dom 𝑅
21eqimss2i 4044 . 2 dom 𝑅𝐴
3 cotr3.b . . . 4 𝐵 = (𝐴𝐶)
4 cotr3.c . . . . 5 𝐶 = ran 𝑅
51, 4ineq12i 4211 . . . 4 (𝐴𝐶) = (dom 𝑅 ∩ ran 𝑅)
63, 5eqtri 2761 . . 3 𝐵 = (dom 𝑅 ∩ ran 𝑅)
76eqimss2i 4044 . 2 (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵
84eqimss2i 4044 . 2 ran 𝑅𝐶
92, 7, 8cotr2 14924 1 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝐴𝑦𝐵𝑧𝐶 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wral 3062  cin 3948  wss 3949   class class class wbr 5149  dom cdm 5677  ran crn 5678  ccom 5681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688
This theorem is referenced by: (None)
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